The document discusses column stability and Euler's formula for pin-ended columns. It provides:
- An introduction to column stability and buckling under compressive loads.
- A model of a column as two rods with a torsional spring to explain buckling behavior.
- Derivation of Euler's formula which gives the critical buckling load of a pin-ended column as a function of its properties.
- Discussion of how column geometry, specifically its slenderness ratio, affects buckling behavior.
Solution Manual for Structural Analysis 6th SI by Aslam Kassimaliphysicsbook
https://www.unihelp.xyz/solution-manual-structural-analysis-kassimali/
Solution Manual for Structural Analysis - 6th Edition SI Edition
Author(s): Aslam Kassimali
Solution Manual for 6th SI Edition (above Image) is provided officially. It include all chapters of textbook (chapters 2 to 17) plus appendixes B, C, D.
This book is intended to cover the basic Strength of Materials of the first
two years of an engineering degree or diploma course ; it does not attempt
to deal with the more specialized topics which usually comprise the final
year of such courses.
The work has been confined to the mathematical aspect of the subject
and no descriptive matter relating to design or materials testing has been
included.
Solution Manual for Structural Analysis 6th SI by Aslam Kassimaliphysicsbook
https://www.unihelp.xyz/solution-manual-structural-analysis-kassimali/
Solution Manual for Structural Analysis - 6th Edition SI Edition
Author(s): Aslam Kassimali
Solution Manual for 6th SI Edition (above Image) is provided officially. It include all chapters of textbook (chapters 2 to 17) plus appendixes B, C, D.
This book is intended to cover the basic Strength of Materials of the first
two years of an engineering degree or diploma course ; it does not attempt
to deal with the more specialized topics which usually comprise the final
year of such courses.
The work has been confined to the mathematical aspect of the subject
and no descriptive matter relating to design or materials testing has been
included.
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
3. • In the design of columns, cross-sectional area is
selected such that
- allowable stress is not exceeded
all
A
P
- deformation falls within specifications
spec
AE
PL
• After these design calculations, may discover
that the column is unstable under loading and
that it suddenly becomes sharply curved or
buckles.
Stability of Structures
4. • Long slender members subjected to axial
compressive force are called columns.
• The lateral deflection that occurs is
called buckling.
• The maximum axial load a column can
support when it is on the verge of
buckling is called the critical load, Pcr.
Stability of Structures
5. • Consider model with two rods and torsional
spring. After a small perturbation,
moment
ing
destabiliz
2
sin
2
moment
restoring
2
L
P
L
P
K
• Column is stable (tends to return to
aligned orientation) if
L
K
P
P
K
L
P
cr
4
2
2
Stability of Structures
6. • Assume that a load P is applied. After a
perturbation, the system settles to a new
equilibrium configuration at a finite
deflection angle.
sin
4
2
sin
2
cr
P
P
K
PL
K
L
P
• Noting that sinθ < θ, the assumed
configuration is only possible if P > Pcr.
Stability of Structures
7. • Summing moments, M = Py, equation of
bending becomes
• General solution is
• Since y = 0 at x = 0, then B = 0.
Since y = 0 at x = L, then
0
2
2
y
EI
P
dx
y
d
x
EI
P
cos
B
x
EI
P
sin
A
y
0
L
EI
P
sin
A
Euler’s Formula for Pin-Ended Beams
8. • Disregarding trivial solution for A = 0, we get
• Which is satisfied if
or
,...
3
,
2
,
1
2
2
2
n
L
EI
n
P
n
L
EI
P
0
sin
L
EI
P
Euler’s Formula for Pin-Ended Beams
9. • Smallest value of P is obtained for n = 1, so critical load for column
is
• This load is also referred to
as the Euler load. The
corresponding buckled
shape is defined by
• A represents maximum
deflection, ymax, which occurs
at midpoint of the column.
2
2
L
EI
Pcr
L
x
sin
A
y
Euler’s Formula for Pin-Ended Beams
10. • A column will buckle about the principal axis of the x-
section having the least moment of inertia (weakest axis).
• For example, the meter stick shown will
buckle about the a-a axis and not
the b-b axis.
• Thus, circular tubes made excellent
columns, and square tube or those
shapes having Ix ≈ Iy are selected
for columns.
Euler’s Formula for Pin-Ended Beams
11. • Buckling equation for a pin-supported long slender column,
Pcr = critical or maximum axial load on column just before it
begins to buckle.
E = modulus of elasticity of material
I = Least modulus of inertia for column’s x-sectional area.
L = unsupported length of pinned-end columns.
2
2
L
EI
Pcr
Euler’s Formula for Pin-Ended Beams
12. • The value of stress corresponding to the critical load,
A
L
EI
A
P
L
EI
P
cr
cr
cr
cr
2
2
2
2
Slenderness Ratio
13. Expressing I = Ar2 where A is x-sectional area of column and
r is the radius of gyration of x-sectional area.
cr = critical stress, an average stress in column just before
the column buckles.
E = modulus of elasticity of material
L = unsupported length of pinned-end columns.
r = smallest radius of gyration of column
2
2
r
L
E
cr
Slenderness Ratio
14. • The geometric ratio L/r in the equation is known as the
slenderness ratio.
• It is a measure of the column’s flexibility and will be used to
classify columns as long, intermediate or short.
Slenderness Ratio
15. IMPORTANT
• Columns are long slender members that are subjected to
axial loads.
• Critical load is the maximum axial load that a column can
support when it is on the verge of buckling.
• This loading represents a case of neutral equilibrium.
Ideal Column with Pin Supports
16. IMPORTANT
• An ideal column is initially perfectly straight, made of
homogeneous material, and the load is applied through the
centroid of the x-section.
• A pin-connected column will buckle about the principal axis
of the x-section having the least moment of intertia.
• The slenderness ratio L/r, where r is the smallest radius of
gyration of x-section. Buckling will occur about the axis
where this ratio gives the greatest value.
• Preceding analysis is limited to centric loadings.
Ideal Column with Pin Supports
20. The A-36 steel W20046 member shown is to be used as a
pin-connected column. Determine the largest axial load it
can support before it either begins to
buckle or the steel yields.
Example
21. From calculation, column’s x-sectional area and
moments of inertia are A = 5890 mm2,
Ix = 45.5106 mm4,and Iy = 15.3106 mm4.
By inspection, buckling will occur about the y-y axis.
Applying buckling equation, we have
kN
6
.
1887
m
4
mm
1000
/
m
1
mm
10
3
.
15
kN/m
10
200
2
4
4
4
2
6
2
2
2
L
EI
Pcr
Example
22. When fully loaded, average compressive stress in column is
Since this stress exceeds yield stress (250 N/mm2), the load P
is determined from simple compression:
kN
5
.
1472
mm
5890
N/mm
250 2
2
P
P
2
2
N/mm
5
.
320
mm
5890
N/kN
1000
kN
6
.
1887
A
Pcr
cr
Example
32. Exercise 1
Determine
(a) The crtical load for the steel strut
(b) The dimension d for which the aluminium strut will have same critical load
(c) Express the weight of the aluminium alloy strut as a percent of the weight of
the steel strut.
Refer to Figure No
10.12 Page 621
Steel
E = 200 GPa
Ρ = 7860 kg/m3
33. Exercise 2
A compression member of 1.5 m effective length consists of a solid 30 mm
diameter brass rod. In order to reduce the weight of the member by 25%, the
solid rod is replaced by a hollow rod of the cross section shown. Determine
(a) the percent reduction in the critical load
(b) The value of the critical load for the hollow rod. Use E = 70 GPa
Refer to Figure No
10.13 Page 621
15
mm
30 mm
30 mm